XK-TST 2022 Problem 4

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben's numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana's numbers and he gets $d$ different integers.

For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben.

a) Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ b) Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ Proposed by Besfort Shala, Kosovo